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This study introduces marginal maximum likelihood (ML) estimation for hierarchical multinomial processing tree (MPT) models. Adaptive Gauss-Hermite quadrature (AGHQ) is recommended for its accuracy and reliability in parameter estimation.

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Area of Science:

  • Cognitive psychology
  • Psychometrics
  • Statistical modeling

Background:

  • Hierarchical multinomial processing tree (MPT) models are widely used in cognitive psychology.
  • Estimating parameters in these complex models, especially with random effects and covariates, presents computational challenges.

Purpose of the Study:

  • To propose and evaluate marginal maximum likelihood (ML) estimation methods for hierarchical MPT models.
  • To compare the performance of three numerical integration methods: Laplace approximation (LA), adaptive Gauss-Hermite quadrature (AGHQ), and Quasi Monte Carlo (QMC).
  • To introduce ML-based model comparison and goodness-of-fit testing for these models.

Main Methods:

  • Development of marginal maximum likelihood (ML) estimation for hierarchical MPT models with random and fixed effects.
  • Implementation and comparison of Laplace approximation (LA), adaptive Gauss-Hermite quadrature (AGHQ), and Quasi Monte Carlo (QMC) for approximating intractable integrals in the likelihood function.
  • Simulation study to assess the bias and coverage rates of the estimation methods.

Main Results:

  • Adaptive Gauss-Hermite quadrature (AGHQ) demonstrated good performance regarding bias and coverage.
  • Quasi Monte Carlo (QMC) integration also performed well, particularly with a sufficient number of responses per participant.
  • Laplace approximation (LA) frequently failed due to undefined standard errors.

Conclusions:

  • AGHQ is a reliable and accurate method for estimating parameters in hierarchical MPT models.
  • QMC is a viable alternative when sufficient data is available.
  • The proposed ML-based methods offer robust tools for model evaluation and comparison in cognitive modeling research.